supplementary angles

cos(t)=sin(t+90)
To solve, it looks like it is written as a weird version as cos(x+y) With x equaling -10, because cos(-10) =cos(10) and sin(190)=sin(-10)

Pretest Questions

I sort of dont know how tp simplify for #10. I am also not sure how to do #16 or 17. For #16, b and c, I dont know how to express the equation. Thanks

supplementary angles

Im having trouble figuring out problems with supplementary angles. I know that if you have cos^2(x) + sin^2(y) where x+y = 180 then cos^2(x) + sin^2(y) = 1. But what is sin(t) in terms of cosine and vice versa? What I mean is sin(t) = cos(?). I was looking back on a problem, how would I solve cos(10)sin(70) + sin(20)sin(190)?

amp, period, vertical shift, formula

f(t)= Asin(bt+c)+d
A= Amplitude
2pi/b= Period
-c/b= Phase Shift
d= vertical shift

number 6b on the pretest

would another set of polar coordinates for the same point just be 4cos(9pi/4 + 2pi) , 4sin(9pi/4+2pi)?

I missed class on friday...

and there are a few things about the 12R book problems and the semester review problems that confused me.

for 12R #s 36 and 43, I think we need to use induction, and I was wondering if Mr. Karafiol said anything about whether we'll need to know this for the final.

as for the semester review problems... in # 60, we are to find dx/dt, but I don't understand what the "d" means. It's definitely not the difference quotient because there is no f(x).

For #65 I completely spazzed on how to find the limit of the series sum if you know the limit of terms within it. I know we just did this stuff in class so I feel really lame, but just because we know that the largest term approaches 4, how do we know what the series will converge to if we don't know how many terms we're adding?

Finally, for the last problem (the second #66) I determined that the common ratio of the series is sec(t)/2 and I know that a geometric series will only reach a finite number if the ratio is between 0 and 1. So I set sec(t) equal to both 0 and 1, and I found a maximum value for t (5pi/3), but the one equal to 0 has no solution. So how do you find the minimum value for t?

thanks.

Angular and Linear Velocity

Can someone give a quick explanation of how you do these calculations. For example, number 6 on the calculator section. For angular velocity i said that it was 108000000*pi/224.7*24 but i am not sure if that's right. Also, how do you convert that to linear velocity?
Thanks

gsp portion

is question 18 on the pretest supposed to relate to the gsp portion of the final?

Properties of Cosines and Sines

Can someone give an explanation or list of the properties of cos and sin because I am having a hard time remembering them. Thanks!

Happy Birthday!

Happy Birthday Abraham de Moivre, I'm so sorry that I didn't celebrate your birthday until this year!

A few snippets from Wikipedia, sorry that I didn't write them myself:

Born May 26, 1667 in Vitry-le-François, Champagne, France, Died November 27, 1754 in London, England

de Moivre was a Calvinist. He left France after the revocation of the Edict of Nantes (1685) and spent the remainder of his life in England.

It is reported in all seriousness that De Moivre correctly predicted the day of his own death. Noting that he was sleeping 15 minutes longer each day, De Moivre surmised that he would die on the day he would sleep for 24 hours. A simple mathematical calculation quickly yielded the date, November 27, 1754. He did indeed pass away on that day. (Yet another factoid you can use on next year's precalc class, Mr. Karafiol, and I'm sorry if you said this to us and I forgot).

He first discovered the "closed form" expression for Fibonacci numbers linking the nth power of phi to the nth Fibonacci number.

http://en.wikipedia.org/wiki/Abraham_de_Moivre

In honor of his 340th birthday, I'm going to recap all the fun stuff you can do with de Moivre's theorem, I'm so sorry if this isn't enough:

First of all, de Moivre's theorem states that (in complex form): r(cos(θ)+isin(θ))ⁿ=rⁿ(cos(nθ)+isin(nθ))
In polar form, it becomes
(r,θ)ⁿ=(rⁿ,nθ)

But remember, it also works backwards to find roots of polar coordinates!
So,
*nrt(r,θ)=(nrt(r),((θ+2kπ)/n)
Remember the +2kπ !

*nrt=nth root, unfortunately I can't write it in pretty print, sorry!

I'm sorry if I forgot anything else you can do with de Moivre's theorem.

Oh, it's also National sorry day in Australia, sorry I didn't mention that at the beginning.

Vector Physics

Hey, does anyone remember how to to the comp of a vector and the proj of a vector onto another? And what exactly does "comp" tell you?

12.5 #19

I understand why it works, just don't understan how i would prove it.

Eccentricity

I forgot how to find the eccentricities of narrow vs. wide ellipses vs. circles. I know wide ellipses have an eccentricity close to 1, circles have an eccentricity of 0 (since focal length is 0), but then what are narrow ellipses? I seem to remember eccentricity must be greater than 0, so is it also between 0 and 1 for narrow ellipses?

Also, for number 8 on the pretest, I'm coming up with y=-15/2, since x^2+16=32 is the same as x^2=32(y-1/2), so 4p=32, and p=8, so the directrix of the parabola would be y=1/2-8=-15/2, which isn't an option. Where did I go wrong, or is the 16 a mistake on the test?

Problem on pretest

when do you use B^2 - 4AC is less than 0, greater than 0, or equal to 0, and what do A, B, and C stand for?

12.4.42 b

I think i am confused by the wording, but also for part a, don't you get (1^4, 4*theta) for an answer? This is not what I got for 41. Any approaches?

A Little About Parametrics for Conics

Ellipses

We know that the standard form for this conic section is (x^2)/a +(y^2)/b = 1

The parametric equation for an ellipse is x = a * cos (t)
y = b * sin (t)

Hyberbolas

We know that the standard form for this conic section is (x^2)/a - (y^2)/b = 1

The parametric equation for an hyperbola is

y = a * sec (t)

x = b * tan (t)

*the y values and x values will vary according to what axis the hyperbola is on

Parabola

The parametric equation for the parabola is

x = t^2/4p

y = t

The standard equation for the parabola is x^2 = 4py and y^2 = 4px

* Make sure to set the mode of the TI-89 to Parametric

parametric equations of conics?

I'm not sure if this will be on the test at all, but to add to everyone's conic charts, here are the parametric equations for all the conic sections:
Ellipse: x = a*cos(t), y=b*sin(t)
Hyperbola: x=a*sec(t) (or a/cos(t)), y=b*tan(t) (or the other way around if the vertices lie on the y-axis)
Parabola: x=t^2/4p, y=t

i hope this is actually helpful...
and speaking of parametrics, does anyone have an idea of the other main concepts that are likely to be on the test regarding parametrics other than the projectile motion equations or expressing parametrics in terms of x and y? (do we need to know about cycloids?)

Degrees of Rotation

I don't believe we're covering degrees of rotation.

Pretest Number 11

3(-)= pi/2 +2k*pi ... so (-)=pi/6 +2/3k*pi. The values are pi/6, 5pi/6, and 9pi/ 6. You can also think of it geometrically, it is going to make an equilateral triangle, so split the points up 3 ways.

Pretest number 11

So I know that sin can get to one and this means that the graph is furthest from pole when 3(-)
=pi/2. So then I know that (-) must be pi/6 and inside 0 to 2pi isnt that the only value for theta? I know it is not so please help me out thank you.

ps (-) is a theta symbol.

Degrees of Rotation

I think I missed the day Mr. Karafiol explained degrees of rotation of conic sections? I read the part in the book, but I'm still kinda confused about it. Can anyone try to explain it to me?

thanks.

Conics Review

Here's a link to a spreadsheet I made to help remember the properties of different conics.

https://www.cjhghs.com/051307 Conics Chart.xls

Pretest #12

Based on the picture i said that the focus was (0,p) and the latus rectum intersected the parabola at (x,p) and (-x,p). I then said the length was 2x and solved x^2=4py for 2x in terms of p. Is that what is supposed to be done? It seems like it should be more complicated.

Semester Review Packet # 54

I know we've done multiple problems like this one but could someone please remind me how to do it?

sin a = 4/5 and sin b = 5/13

find sin (a-b)

Recap of Conics

In light of the test on monday, I'm taking it upon myself to recap some important values and other stuff for conics
Ellipses:
Vertex=the point where the major axis and the ellipse intersect
a=length of the semimajor axis (half the larger axis)=vertex-center
b=length of the semiminor axis (half the shorter axis)
c=distance from focus to center, c^2=a^2-b^2
The foci are always on the semimajor axis
general equation:
(x-h)^2/a^2+(y-k)^2/b^2=1 for horizotnal major axis
(y-k)^2/a^2+(x-h)^2/b^2=1 for vertical major axis

Parabolas:
Vertex= the point at the bottom of the parabola (top if the parabola is upside down), x-coord of the vertex=-b/2a if ax^2+bx+c=0
p: for an equation like this: (x-h)^2=a(y-k), a=4p, p represents the distance from the vertex to the focus or directrix

Hyperbolas:
a=distance from the center to the vertex of one curve of the hyperbola
b: the equation of the asymptote is (y-k)=b/a(x-h)
c: distance from the center to either focus, c^2=a^2+b^2 (remember, plus in hyperbolas, minus in ellipses)
genral formula:
(x-h)^2/a^2-(y-k)^2/b^2=1 for hyperbolas with horizontal axes,
(y-k)^2/a^2-(x-h)^2/b^2=1 for vertical hyperbolas

If you want to add anything, please do! I want to know what I've forgotten.

-Kevin

Hint!

Dont worry I will make a serious blog post but I thought this was funny. In the book (as I am sure you noticed) under problem 12.4.39 it says [Hint:2 = 1 + 1]

Missing Problems

I've noticed that problems 31-39 are missing on the semester review packet. If he sent them via e-mail to people through something that I guess I'm not a member to could someone please pass that along?

my e-mail is brianh91@aol.com

#59

For number 59 what type of equation are we basing the Fibonacci sequences on? Thanks.

Sums

What is a partial sum?? Can someone explain that to me? Also, how do u do a summation on your calculator. And can someone briefly sum up what happened on Tuesday because I missed most of block. Thanks.

I'm So Lost

I need a lot of help on Chapter 12 because once again I was not at school. Can some one fill me in on the new calculator operations that the email was discussing? Also I have questions about how to convert equations like this: 3x^2 + 4y^2 + 2x + y + 22 = 0, to ellipse, hyperbola, or porabola form.

Sum feature on calculator

Can someone teach me how to use the sum feature for a series on the calculator?
And the multiplication one for geometric series too... thanks

sketchpad

hey david, about that sketchpad assignment, this is the only way i had to contact you about doing the sketchpad for tuesday (i won't be in class on monday either, another ap test). so, just send me an email at elleinad579@gmail.com so we can discuss ways to work about.

(sorry about the irrelevant post guys.)

12.1.26

I wrote this as a recursive series. an=an-1 -3 +10*-1^(n+1) With a0=1.

12.1.61

In problems where you have a Asubn I.E.: 5(an)^2+4(-1)^n and you are not given any values for an how do you figure out what to use for it? Is it just the value of n or what?

12.1.26

I missed class on friday, and although i read the book to clarify 12.1-12.3, there're still a couple things i'm not sure about.
For this question, I can see the pattern of the terms. Every other interval has a difference of -13, and the intervals inbetween those have a difference of 7. But how do I convert this pattern into an equation for a[n], since there's no constant difference? Did you learn a trick in class, or is this guess and check?

Explanation of 12.3.37

Problem number 37 is the question about the ball drop, which I did the hard way until I looked back at the chapter. You do the exact same calculation that you would for a normal geometric series, with a1(1-r^k)/(1-r), except that you have to add the initial drop distance, and calculate a1. If an represents the distance from the ground, up, and back to the ground again, then a1 would be 8, because you go up 4 feet and then fall down 4 feet.

What did we go over in class?

For me and all those other people who were taking the APUSH Test, what did we go over in class? I realize we're starting a new chapter, so I don't want to be too far behind, since I'm also going to miss tomorrow's class for the AP Bio test. Anything really important that I should know?

So, I feel like I'm not contributing anything, so, here's how to get 29 in 12.2:

You may have gone over this in class, but the way to think about arithmetic sequences is to think about what you get when you add the first and last terms. This is the same as what you get when you add the second and second-to-last terms, and so on. So you multiply the sum of the first and last terms by n, then divide by two. So, 2+100=102, there are 50 ((100-2)/2+1) terms (to get the number of terms you subtract the first term from the last term and divide by the common difference, then add one). You get 510, so divide by two to get 255. Yay!

-Kevin

12.1.26, 59c, & 61

12.1.26
for this problem, i noticed that they alternate changing by 13 and -7, the signs change, and a1=8 but i didn't know how to put that into an equation. all i know is that 8*(-1)^n is in there. how do i finish it?

12.1.59c
the book gave an explanation for the pattern but it didn't make any sense to me. help?

12.1.60
is there a typo? does it mean to say integers less than one or equal to 15? and if so, what is the question asking because that's confusing too

thanks, monica

12.3#37

This was the problem with the ball that was dropped and rose halfway after each bounce. I figured out the answer by doing the sum by hand, but couldn't figure out a formula. Any hints?

Notes from Class

Is there anyway that someone could post the class notes from Friday, May 11 on the blog? I was sick so I wasn't able to get them.

10-Review #49

I'm having trouble remembering how to do this one. It asks how to rotate the x-y axes to eliminate xy from x^2+xy+y^2-3y-6=0. Any help?

Friday

I was away at State Science Fair can someone fill me in on what we did on Friday?

10.3

I have problems with #61, i just cant picture it right. please tell me what it the problem is really talking about.
I also dont know how to start to prove the problem on #55.
thanks
mel

Projectile Problems

Can someone explain to me how to do the projectile motion problems again?

10.3.A 9&11

I somehow managed to get the answers switched around for 9 and 11: the answer i got for 9 is listed in the textbook as the answer to 11 and vice versa. For 11, i got cot(2*theta) = (17-31)/48, which simplifies to -7/25. plugging -7/25 into the half-angle identities, I got that sine of theta is the square root of (1- -7/25)/2, which comes out to 4/5, making the cosine 3/5. but the answer in the book has the sine and cosine values switched around. Same goes for number 9, which has the cotangent of the double angle equal to positive 7/25. can someone tell me what i'm doing wrong (assuming the book isn't wrong)?

#41

I don't understand how you're supposed to factor these equations. Are we supposed to answer these questions without having to factor out?

Parabola Equations

The equation is y^2=4px, or x^2=4py depending on the equation. A parabola is a set of points that are equidistant from the directorix and the focus. The p is the distance from the vertex of the parabola to the focus, also the distance from the vertex to the directorix. To find p, solve for x^2 in the equation, and then set it equal to 4py. You then move p units away from the vertex to find the directorix or focus.

Questions like 10.3 #41...

There are a few of these where we are meant to simplify equations like 41x^2-24xy+34y^2-25=0 and then graph them. It's possible to do this if you pretend the equation is all one big quadratic in x or y (for example, if this one was in the form Ax^2+Bx+C=0, A=41, B=-24y, and C=(34y^2-25)). However, it's easiest to just solve them on the calculator for y and then graph the resulting solutions in function mode (there should always be two values for y, so graph both). #s 41-49odd are all either ellipses, parabolas, or hyperbolas, and they are all rotated. However, I had trouble figuring out what the shapes were until I graphed them. CJ explained some sort of trick below for determining the shape before graphing, but I don't really understand how finding the discriminant tells you the shape... does anyone have a different explanation?

10.3 #35 Discriminant Help

So for number 35 is the discriminant would be found by taking A as the coefficient for x^2 B as the coef for y^2 and C as the coef for x? Does that always stay constant or do you just take the coefficients in order. IE: First coefficient in the order of terms is A second is B etc...

Finding the Shape of a Curve from the Discriminant

So apparently there's this trick to finding out the shape of a curve from the discriminant of it's equation. So let's say you have the equation that starts with 2x^2+2xy+4y^2...followed by some other stuff. Knowing that the discriminant is b^2-4ac, that gives us 2^2-4*2*4, which equals -28. Since the result is negative, the shape must be an ellipse. If it had been equal to 0, it would have been a parabola, and if it had been greater than 0, it would have been an ellipse. I don't think we learned this in class (unless I was dreaming about my girlfriend), but it's on the homework for tomorrow.

parabolas

how do you find the equation for a parabola. is it y^2=4px? what is the p? i dont understand! also how do you find the focus and directrix?

Monday's Homework (Response to Taylor)

10.2: 59, 61
10.3: 35-49 odd, 53, 55, 56, 57, 60
10.3.A: 1-13 odd

I remember him saying we didn't have to do something but I'm not sure it its 10.3.A so I put it anyways.

monday hw

can anyone tell me what our homework is for monday? i lost my assignment sheet and it's not uploaded on mr. karafiol's site, i do remember though that we were supposed to skip section 10.3A.

thanks.

10.2#61

I'm not sure how to finsih this problem. I uunderstand that the punch bowl and the cups are the two foci of an ellipse and that the sum of the two distances from both foci to any point on the ellipse is 100. How do i use this to find the circumference of the ellipse?